Spheres and Cones

April 19, 2022

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Calculating areas and volumes of spheres and cones

The formulae that are given to you in your exam are shown below:

The proofs for these formulae are beyond the scope of this mathematics course so we will only need to focus on actually using them rather than where they come from.

Using the correct formula we just need to put each known value into the formula to get what we need. As well as this we can use the formula in reverse; so, if we were asked to work out a radius using the volume of a cone, for example, this can easily be done.

Example

If we have a shape with a radius of 8cm, find the following to 2dp:

1) The surface area of a sphere

2) The curved surface area of a cone with length 6cm

3) The volume of a sphere

4) The volume of a cone with height 4cm

1) Here we need to use the equation $4 \pi r^2$ with a radius of 8cm gives us a value of $4 \times \pi \times 8^2=804.25cm^2$

2) Using the equation $\pi \times r \times l$ we get $8 \times 6 \times\pi =150.80cm^2$

3) The equation for the volume of a sphere is $\frac{3}{4}\pi r^3$ and we have a radius of 8cm so we get $\frac{4}{3} \times8^3 \times\pi = 214.66cm^2$

4) A volume of a cone is $\frac{1}{3} \pi r^2 h$ which gives $\frac{1}{3} \times 8^2 \times 4 \times\pi =268.08cm^2 4$

Using the formulae that are shown above we simply need to be careful to add the correct values for the radius, height and lengths to be able to find the correct value for what we want.

Constructing the graphs of circular loci

We will at times need to draw a curve that stays at a fixed distance from a certain point; this will clearly be a circle with the point at the middle. These come in very handy when we are trying to draw certain graphs of the type where r is the radius of the circle. This is clearly going to be fairly easy to draw on a graph as we simply set the radius of our compass to the required value for r and then draw a circle with the origin (0, 0) as the centre.